However, once a non-degenerate prior \(\pi \)  is employed, \(C(n,t,s)\)  depends on \(n\)  and, as we show in the next section, converges to 0 as \(n\rightarrow \infty \)  under mild assumptions on \(\pi \) . Thus, \(\liminf R(n,t,s)>0\)  is not everymore sufficient to establish whether consistency holds true or not.
Instead, one needs to compare the rate at which \(C(n,t,s)\)  converges to 0 with the behaviour of \(R(n,t,s)\) , as done in the following sections.
Note that further lower bounds for \(R(n,t,s)\)  for general values of \(s\)  are given in [1]}, [2]}.
However, once combined with \(C(n,t,s)\) , these are too loose to deduce either consistency or lack thereof. Therefore, we need to exploit different techniques to determine the rate of \(R(n,t,s)\) .
Since \(\text{pr}( K_n = t \mid X_{1:n}) = \int \text{pr}( K_n = t \mid X_{1:n}, \alpha ) \pi (\alpha \mid X_{1:n}) \, \mathrm {d}\alpha \) , by (REF ) we deduce \(\lim \sup \, \text{pr}( K_n = t \mid X_{1:n},\alpha ) <1\)  for every \(\alpha >0\) .
This, however, does not imply that \(\lim \sup \text{pr}( K_n = t \mid X_{1:n})<1\) , as one first needs to ascertain whether limit and integral can be interchanged. The main reason is that, in the asymptotic regime we are considering, the posterior distribution \(\pi (\alpha \mid X_{1:n})\)  concentrates around 0 as \(n\rightarrow \infty \) , see Proposition REF  above.
